Write the formula for a logistic function that has values betw...

Question

Write the formula for a logistic function that has values between y = 0 and y = 1, crosses the line y = 1/2 at x = 0, and has slope 5 at this point.

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find a logest function with an output between 0 and 1 that crosses Y is equal to 1/2 at x is equal to 0. And has a slope of -8 at that point. OK, so. Essentially, it appears we're asked to solve a log-based function based on all the information that is provided to us by the pro cell. So we're ultimately trying to determine what this specific function that is log-based will be. So, now that we know what we're ultimately trying to solve for, our first step that we need to take is we need to determine what the general form of this log-based function will be. So, as we should recall, a common log-based function will be Y is equal to L divided by 1 + E to the power of K multiplied by parentheses of X minus X subscript 0. Where L is the maximum value, and let us note that the maximum value here will mean that L is equal to 1 for this particular case. K represents the slope, specifically, K controls the slope. X subscript 0 is the midpoint, and the value of X where Y is equal to L divided by 2, which will be equal to 1/2. Fantastic. Our second step now is we need to recall and use the crossing condition at X is equal to 0. So, as we should note, and based on our prior knowledge. And since we'll need to note that we want the function to cross at, we want it to cross at y is equal to 1/2, at x is equal to 0, so that means we need to write 1 divided by 1 plus e to the power of k multiplied by parentheses of 0 minus x subscript 0. And this will be all equal to 1/2. Which will mean this simplifies to, we can simplify this to write 1 divided by 1 plus e to the power of k multiplied by x subscript 0 is equal to 1/2. Fantastic. So now we need to perform cross multiplication. We need to cross-multiply, and when we do, we will find that we could write 2 is equal to 1 plus. E to the power of K multiplied by x subscript 0. Which will mean that E to the power of K multiplied by X subscript 0 will be equal to 1, and when we take the natural logarithm, we will find that K multiplied by X subscript 0 will be equal to 0, which will mean that X subscript 0 will be equal to 0. Fantastic. So that will mean that our log-based function will simplify to Y is equal to 1 divided by 1 + E to the power of K multiplied by X. Fantastic. So now for our 3rd step is we need to find the slope M, so the slope M. At X is equal to 0. So now we need to find the slope at x is equal to 0. And in order to do that, we need to focus our attention on the derivative of this log-based function, so we need to find y is equal to d by dx 1 divided by 1 plus e to the power of k. X, and we need to use the chain rule. So Y is equal to e to the power of K multiplied by X multiplied by K divided by parentheses of 1 plus e to the power of K multiplied by X to the power of 2. And when we are at x is equal to 0, that will mean that Y 0 will be equal to e 0 multiplied by k divided by parentheses of 1 plus e to the power of 0 to 2. Which will be equal to when we simplify our final answer down to its most simple form, K divided by 4. However, we want to note that Y 0 is equal to -8 as it states by the prom itself. So that will mean therefore that K divided by 4 will be equal to -8. Which will mean without a doubt that hey. Will be once again. We need to note that K, so let's take a moment here to pause, we could safely say that K will be equal to -32. So that means our final log-based function will be, so our final answer will be, our function Y will be equal to 1 divided by 1 plus e to the power of 32 X. And that's it. That is our final answer. Hooray, we did it. So going back to look at the prom itself, we could safely say once again that our final answer is Y is equal to 1 divided by 1 + e to the power of 32 X. And that's it, that is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Write the formula for a logistic function that has values between y = 0 and y = 1, crosses the line y = 1/2 at x = 0, and has slope 5 at this point.

Watch next